1. ## ANother model

Hi people I dont know if anyone can help, but I'm working on some models and I'm having a bit of difficulty with this one..

$\displaystyle \dot S(t) \equiv \frac{{dS}} {{dt}} = - \alpha SI$

where is alpha is a const.

rate which no. of those infected changes depending on suseptibles getting ill and other recovering,

This process is described by

$\displaystyle \dot I(t) \equiv \frac{{dI}} {{dt}} = \alpha SI - \beta {\rm I}$

where beta is a const.

So i need to know how to form the differential equation for I(S) and given that I=1 when S=N, find solution for I(S).

2. Originally Posted by reivera
Hi people I dont know if anyone can help, but I'm working on some models and I'm having a bit of difficulty with this one..

$\displaystyle \dot S(t) \equiv \frac{{dS}} {{dt}} = - \alpha SI$

where is alpha is a const.

rate which no. of those infected changes depending on suseptibles getting ill and other recovering,

This process is described by

$\displaystyle \dot I(t) \equiv \frac{{dI}} {{dt}} = \alpha SI - \beta {\rm I}$

where beta is a const.

So i need to know how to form the differential equation for I(S) and given that I=1 when S=N, find solution for I(S).
$\displaystyle \frac{dI}{dS} = \frac{dI}{dt} \, \cdot \frac{dt}{dS} = (\alpha S I - \beta I) \, \left( -\frac{1}{\alpha S I} \right) = \frac{\beta - \alpha S}{\alpha S}, \, I \neq 0,$

$\displaystyle = \frac{\beta}{\alpha} \, \frac{1}{S} - 1$,

subject to the boundary condition $\displaystyle 1 = I(N)$.

I get $\displaystyle I = \frac{\beta}{\alpha} \ln \left( \frac{S}{N} \right) - S + N + 1$ as the solution.

Question for you: This solution is valid provided $\displaystyle I \neq 0$. Is this condition always met?

3. Your answer seems fine to me, I dont think there is any reason I = 0 is a problem, in this model I(t) is the number currently infected...